Perhaps you mean he proved O3 by induction, as you said in the first post. This happens in section 11.8 in my book (An Introduction to Gödel’s Theorems, second edition, 2013). I was initially confused because in Section 11.3, where order-adequacy properties are introduced, the author says that their proofs are boring and does not seems to include them.
Now induction in the proof of O3 is meta-theoretical rather than a part of Robinson's arithmetic. That is, this induction is a part of the reasoning we use to prove claims about derivability of formulas in Q. You may note that the author says several times, "Arguing inside Q". This means that the part that follows can be converted into a formal derivation inside Q, and the English text is an outline of that derivation. In such case we can't use induction. But we can, for example, prove the claim $Q\vdash \forall x, y\;S^n x=S^n y\to x = y$ for every fixed natural number $n$ by meta-theoretical induction. That is, we prove that if there exists a derivation of $\forall x, y\;S^n x=S^n y\to x = y$, then there exists another derivation of $\forall x, y\;S^{n+1} x=S^{n+1} y\to x = y$. Neither derivation includes the axiom of induction because it is not listed in axioms of Q in Section 10.3.
One can prove O3 without invoking induction, but then one has to say something like "... and so on", so the proof becomes less precise. Similarly, I guess, one can use meta-theoretical induction in the proof of O2 because one has to construct a derivation that considers $n+1$ different cases $x=\bar{0},\ldots,x=\bar{n}$, and without induction one has to say, "continuing like this, ..." or something similar. The important point is that the resulting formal derivation does not contain the induction axiom.
In the future, please refer to precise places in the book that you are asking about.
